What is the general solution for |sin(x)|=sqrt(a) ?

The general solution for |sin(x)|=sqrt(a) is No Solution for x\in\mathbb{R}

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

|sin(x)|=sqrt(a)

Solution

∣ sin (x) ∣ = a

Solution

No Solution for x \in R

Solution steps

∣ sin (x) ∣ = a

Solve by substitution

∣ sin (x) ∣ = a

Let:

sin (x) = u

∣ u ∣ = a

∣ u ∣ = a :

No Solution for

u \in R

∣ u ∣ = a

Find positive and negative intervals

Find intervals for

∣ u ∣

u \geq 0

u \geq 0, ∣ u ∣ = u

Rewrite

∣ u ∣

for

u \geq 0 : ∣ u ∣ = u

Apply absolute rule: If

u \geq 0

then

∣ u ∣ = u

∣ u ∣ = u

u < 0

u < 0, ∣ u ∣ = - u

Rewrite

∣ u ∣

for

u < 0 : ∣ u ∣ = - u

Apply absolute rule: If

u < 0

then

∣ u ∣ = - u

∣ u ∣ = - u

Identify the intervals:

u < 0, u \geq 0

Domain of

∣ u ∣ = a : a \geq 0

Domain definition

Find non-negative values for radicals:

a \geq 0

∣ u ∣ = a

f (x) \Rightarrow f (x) \geq 0

a \geq 0

The function domain

a \geq 0

Combine intervals with domain

u < 0, u \geq 0

∣ u ∣ u < 0 and a \geq 0 - u \geq 0 and a \geq 0 +

u < 0, u \geq 0

u < 0, u \geq 0

Solve the inequality for each interval

u < 0, u \geq 0

For

u < 0 and a \geq 0 :

No Solution

For

u < 0 and a \geq 0

rewrite

∣ u ∣ = a

- u = a

- u = a : u = - a

- u = a

Divide both sides by

- 1

- u = a

Divide both sides by

- 1

\frac{- u}{- 1} = \frac{a}{- 1}

Simplify

u = - a

u = - a

Combine the intervals

No Solution and u < 0 and a \geq 0

Merge Overlapping Intervals

No Solution and u < 0 and a \geq 0

The intersection of two intervals is the set of numbers which are in both intervals

No Solution

and

u < 0 and a \geq 0

No Solution

No Solution

For

u \geq 0 and a \geq 0 :

No Solution

For

u \geq 0 and a \geq 0

rewrite

∣ u ∣ = a

u = a

u = a : u = a

u = a

Could not simplify further

u = a

Combine the intervals

No Solution and u \geq 0 and a \geq 0

Merge Overlapping Intervals

No Solution and u \geq 0 and a \geq 0

The intersection of two intervals is the set of numbers which are in both intervals

No Solution

and

u \geq 0 and a \geq 0

No Solution

No Solution

Combine Solutions:

No Solution

No Solution for u \in R

Substitute back

u = sin (x)

No Solution for sin (x) \in R

No Solution for sin (x) \in R

sin (x) =

No Solution

: x = arcsin (No Solution) + 2 πn, x = π + arcsin (- No Solution) + 2 πn

sin (x) = No Solution

Apply trig inverse properties

sin (x) = No Solution

General solutions for

sin (x) =

No Solution

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π + arcsin (a) + 2 πn

x = arcsin (No Solution) + 2 πn, x = π + arcsin (- No Solution) + 2 πn

x = arcsin (No Solution) + 2 πn, x = π + arcsin (- No Solution) + 2 πn

Combine all the solutions

x = arcsin (No Solution) + 2 πn, x = π + arcsin (- No Solution) + 2 πn

Since the equation is undefined for:

arcsin (No Solution) + 2 πn, π + arcsin (- No Solution) + 2 πn

No Solution for x \in R

Graph

Popular Examples

-2cos^2(θ)=-1

- 2 cos^{2} (θ) = - 1

sin(xpi)=0

sin (x π) = 0

sqrt(3)tan(4x)+1=0

3 tan (4 x) + 1 = 0

cos(2x-(3pi)/4)+1/(sqrt(2))=0

cos (2 x - \frac{3 π}{4}) + \frac{1}{2} = 0

2sin(x-pi/3)+sqrt(2)=0

2 sin (x - \frac{π}{3}) + 2 = 0

Frequently Asked Questions (FAQ)

What is the general solution for |sin(x)|=sqrt(a) ?
The general solution for |sin(x)|=sqrt(a) is No Solution for x\in\mathbb{R}